Protein segments that form amphipathic alpha-helices in their native state have periodic variation in the hydrophobicity values of the residues along the segment, with a 3.6 residue per cycle period characteristic of the alpha-helix. The assignment of hydrophobicity values to amino acids (hydrophobicity scale) affects the display of periodicity. Thirty-eight published hydrophobicity scales are compared for their ability to identify the characteristic period of alpha-helices, and an optimum scale for this purpose is computed using a new eigenvector method. Two of the published scales are also characterized by eigenvectors. We compare the usual method for detecting periodicity based on the discrete Fourier transform with a method based on a least-squares fit of a harmonic sequence to a sequence of hydrophobicity values. The two become equivalent for very long sequences, but, for shorter sequences with lengths commonly found in alpha-helices, the least-squares procedure gives a more reliable estimate of the period. The analog to the usual Fourier transform power spectrum is the "least-squares power spectrum", the sum of squares accounted for in fitting a sinusoid of given frequency to a sequence of hydrophobicity values. The sum of the spectra of the alpha-helices in our data base peaks at 97.5 degrees, and approximately 50% of the helices can account for this peak. Thus, approximately 50% of the alpha-helices appear to be amphipathic, and, of those that are, the dominant frequency at 97.5 degrees rather than 100 degrees indicates that the helix is slightly more open than previously thought, with the number of residues per turn closer to 3.7 than 3.6. The extra openness is examined in crystallographic data, and is shown to be associated with the C terminus of the helix. The alpha amphipathic index, the key quantity in our analysis, measures the fraction of the total spectral area that is under the 97.5 degrees peak, and is a characteristic of hydrophobicity scales that is consistent for different sets of helices. Our optimized scale maximizes the amphipathic index and has a correlation of 0.85 or higher with nine previously published scales. The most surprising feature of the optimized scale is that arginine tends to behave as if it were hydrophobic; i.e. in the crystallographic data base it has a tendency to be on the hydrophobic face of teh amphipathic helix. Although the scale is optimal only for predicting alpha-amphipathicity, it also ranks high in identifying beta-amphipathicity and in distinguishing interior from exterior residues in a protein.(ABSTRACT TRUNCATED AT 400 WORDS)